Skip to main content
SpringerLink
Log in

Cost Allocation in Combinatorial Optimization Games

  • Chapter
Pareto Optimality, Game Theory And Equilibria

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 17))

Cooperative game theory is concerned primarily with groups of players who coordinate their actions and pool their winnings. One of the main concerns is how to divide the extra earnings (or cost savings) among the members of the coalitions. Thus a number of solution concepts for cooperative games have been proposed. In this chapter, a selection of basic notions and solution concepts for cooperative games are presented and analyzed in detail. The paper is particularly concerned with cost allocation methods in problems that arise from the field of combinatorial (discrete) optimization.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aarts. H.: Minimum Cost Spanning Tree Games and Sets Games. PhD Thesis, University of Twente. The Netherlands (1994)

    Google Scholar 

  2. Aumann, R.J., Maschler. M.: The Bargaining Set for Cooperative Games. In: Dresher, M., Shapley, L.S., Tucker, A. (eds) Advances in Game Theory. Princeton University Press, 443–476 (1964)

    Google Scholar 

  3. Balinski, M.L., Gale, D.: On the Core of the Assignment Game. In: Functional Analysis, Optimization and Mathematical Economics. Oxford University Press, New York, 274–289 (1990)

    Google Scholar 

  4. Bird, C.G.: Cost Allocation for a Spanning Tree. Networks 6, 335–350 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  5. Borm, P., Fiestras–Janeiro, G., Hamers, H., Sanchez, E., Voorneveld, M.: On the Convexity of Games Corresponding to Sequencing Situations with Due Dates. European Journal of Operational Research, 136(3), 616–634 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borm, P., Hamers, H., Hendrickx, R.: Operations Research Games: A Survey TOP. 9, 139–216 (2001)

    MATH  MathSciNet  Google Scholar 

  7. Bodin, L.D., Golden, B.L., Assad, A.A., Ball, M.O.: Routing and Scheduling of Vehicles and Crews. Computers and Operations Research, 10, 63–211 (1983)

    Article  MathSciNet  Google Scholar 

  8. Cachon, G, Netessine, S.: Game Theory in Supply Chain Analysis. In: Simchi-Levi, D., Wu, S.D., Chen, Z.M. (eds) Handbook of Quantitave Supply Chain Analysis: Modeling in the E-Business Area. 13–65, (2004)

    Google Scholar 

  9. Chardaire, P.: Facility Location Optimization and Cooperative Games PhD Thesis, University of East Anglia (1998)

    Google Scholar 

  10. Chardaire, P.: On the Core of the Facility Location Games. Unpublished manuscript available from http://www.sys.uea.ac.uk/people/pc

  11. Claus, A., Kleitman, D.J.: Cost Allocation for a Spanning tree. Networks, 3, 289–304 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  12. Curiel, I., Pederzoli, G., Tijs, S.: Sequencing Games. European Journal of Operational Research, 40, 344–351 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. Curiel, I.: Cooperative Games and Applications, Cooperative Games Arising from Combinatorial Optimization. Kluwer Academic Publishers (1997)

    Google Scholar 

  14. Denavur, N.R., Mihail, M., Vazirani, V.V.: Strategyproof Cost-sharing Mechanisms for Set Cover and Facility Location Games. EC03, June 9–12 (2003)

    Google Scholar 

  15. Deng, X.: Combinatorial Optimization and Coalition Games. In: Du, D.Z., Pardalos, P.M. (eds) Handbook of Combinatorial Optimization. Kluwer Academic Publishers, 77–103 (1998)

    Google Scholar 

  16. Deng, X., Ibaraki, T., Nagamochi, H.: Combinatorial Optimization Games. Proceedings 8th Annual ACM-SIAM Symposium on Discrete Algorithms, 720–729 (1997)

    Google Scholar 

  17. Deng, X., Ibaraki, T., Nagamochi, H.: Algorithmic Aspects of the Core of Combinatorial Optimization Problems. Mathematics of Operations Research, 24, 751–766 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Derigs, U., Grabenbauer, G.: Intime - A New Heuristic Approach to the Vehicle Routing Problem with Time Windows, with a Bakery Fleet Size. American Journal of Mathematical and Management Sciences, 13(3–4), 249–266 (1993)

    MATH  Google Scholar 

  19. Derks, J., Kuipers, J.: On the core of Routing Games, International Journal of Game Theory, 26, 193–205 (1997)

    MATH  MathSciNet  Google Scholar 

  20. Driessen, T.: Cooperative Games, Solutions and Applications. Kluwer Academic Publishers (1988)

    Google Scholar 

  21. Dror, M.: Cost Allocation: The Traveling Salesman, Bin Packing, and the Knapsack. Appl. Math. Comput., 35, 195–207, (1990)

    Article  MathSciNet  Google Scholar 

  22. Dror, M.: Arc Routing Theory, Solutions and Applications. Kluwer Academic Publishers (2000)

    Google Scholar 

  23. Engevall, S.: Cost Allocation in Distribution Planning. Licentiate Thesis, Linkoping Institute of Technology (1996)

    Google Scholar 

  24. Engevall, S.: Cost Allocation in Some Routing Problems - A Game Theoretic Approach. PhD Thesis, Linkoping Institute of Technology (2002)

    Google Scholar 

  25. Faigle, U., Kern, W.: On Some Approximately Balanced Combinatorial Cooperation Games. Methods and Models of Operations Research, 38, 141–152 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  26. Faigle, U., Kern, W., Kuippers, J.: On the Computation of the Nucleolus of a Cooperative Game. International Journal of Game Theory, 30(1), 79–98 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Faigle, U., Fekete, S., Hochstarrler, W., Kern, W.: On the Complexity of Testing Membership in the Core of Min Cost Spanning Tree Games. Int. Journal of Game Theory, 26, 361–366 (1997)

    MATH  Google Scholar 

  28. Faigle, U., Fekete, S., Hochstarrler, W., Kern, W.: The Nukleon of Cooperative Games and an Algorithm for Matching Games. Mathematical Programming, 98, 195–211 (1997)

    Google Scholar 

  29. Faigle, U., Fekete, S., Hochstarrler, W., Kern, W.: On Approximately Fair Cost Allocation in Euclidean TSP Games. OR Spektrum, 20, 29–37 (1998)

    Article  MATH  Google Scholar 

  30. Fekete, S.: Geometric Ideas for Graph Representation and for Cooperative Game Theory. Habilitation Thesis, Mathematisches Institut zu Koln (1997)

    Google Scholar 

  31. Fekete, S., Pulleyblank, W.: A Note on the Traveling Preacher Problem. Available online: http://citeseer.ist.psu.edu/271198.html

  32. Fernandez, F.R., Hinojosa, M.A., Puerto, J.: Multi-Criteria Minimum Cost Spanning Tree Games. European Journal of Operational Research, 158, 399–408 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. Goemans, M.X., Skutella, M.: Cooperative Facility Location Games. Journal of Algorithms, 50, 194–214 (2004)

    Article  MATH  Google Scholar 

  34. G\ddot{o}the-Lundgren, M., J\ddot{o}rnsten, K., V\ddot{a}rbrand, P.: On the Nucleolus of the Basic Vehicle Routing Game. Mathematical Programming, 72, 83–100 (1996)

    Google Scholar 

  35. Granot, D., Granot, F.: Computational Complexity of a Cost Allocation Approach to a Fixed Cost Spanning Forest Problem. Mathematics of Operations Research, 17, 765–780 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  36. Granot, D., Huberman, G.: On the Core and Nucleolus of Minimum Cost Spanning Tree Games. Mathematical Programming, 29, 323–347 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  37. Gutin, G., Punnen, A.: The Traveling Salesman Problem and Its Variations. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  38. Hamers, H.: Sequencing and Delivery Situations: A Game Theoretic Approach. PhD Thesis, Tilburg University,Tilburg , The Netherlands (1995)

    Google Scholar 

  39. Hamers, H.: On the Concavity of Delivery Games. European Journal of Operational Research, 99, 445–458 (1997)

    Article  MATH  Google Scholar 

  40. Hamers, H., Klijn, F., Suijs, J.: On the Balancedness of Multimachine Sequencing Games. European Journal of Operational Research, 119, 678–691 (1999)

    Article  MATH  Google Scholar 

  41. Hamers, H., Borm, P., Leensel, R., Tijs, S.: Cost Allocation in the Chinese Postman Problem. European Journal of Operational Research, 118, 153–163 (1999)

    Article  MATH  Google Scholar 

  42. Hamers, H., Suijs, J., Tijs, S.H., Borm, P.: The Split Core for Sequencing Games. Games and Economic Behavior, 15, 165–176 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  43. Hamers, H., Klijn, F., Solymosi, T., Tijs, S., Vermeulen, D.: On the Nucleolus of Neighbor Games. European Journal of Operational Research, 146, 1–18 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  44. Hartman, B.C., Dror, M., Shaked, M.: Cores of Inventory Centralization Games. Games and Economic Behavior, 31, 26–49 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  45. Hochstattler, W.: Duality in Combinatorial Optimization - Sometimes it Works, Sometimes it Won’t. Available online: http://citeseer.ist.psu.edu/227010.html

  46. Kalai, E., Zemel, E.: Generalized Network Problems Yielding Totally Balanced Games. Operations Research, 30, 998–1008 (1982)

    Article  MATH  Google Scholar 

  47. Kuipers, J.: A Note on the 5-Person Traveling Salesman Game. Mathematical Methods of Operations Research, 38, 131–140 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  48. Kuipers, J., Feltkamp, V., Tijs, S.H.: Vertex Weighted Steiner Tree Games. Report M96-07, Department of Mathematics, Maastricht University (1996)

    Google Scholar 

  49. Kolen, A.: Solving Covering Problems and the Uncapacitated Plant Location Algorithms. European Journal of Operational Research, 12, 266-278 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  50. Lawer, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G., Shmoys, D.B.: The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley and Sons (1985)

    Google Scholar 

  51. Lawer, E.L., Lenstra, J.K., Rinnoy Kan, A.H.G., Shmoys, D.B.: Sequencing and Scheduling: Algorithms and Complexity. In: Graves, S., Rinnooy Kan, A., Zipkin, P. (eds) Logistics of Production and Inventory. North Holland, Amsterdam, 445–522 (1993)

    Chapter  Google Scholar 

  52. Markakis, E., Saberi, A.: On the Core of the Multicommodity Flow Game. Decision Support Systems, 39, 3–10 (2005)

    Article  Google Scholar 

  53. Matsui, T.: A Minimum Taxrate Core Allocation of Bin Packing Problems. Available online: http://citeseer.ist.psu.edu/matsui99minimum.html

  54. Megiddo, N.: Cost Allocation for Steiner Trees. Networks, 8, 1–6 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  55. Megiddo, N.: Computational Complexity of the Game Theory Approach to Cost Allocation for a Tree. Mathematics of Operations Research, 3, 189–196 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  56. Nouweland, A.V.D., Krabbenborg, M., Potters, J.: Flowshop with a Dominant Machine. Europena Journal of Operational Research, 62, 38–46 (1992)

    Article  MATH  Google Scholar 

  57. Nunez, M., Rafels, C.: Characterization of the Extreme Core Allocations of the Assignment Game. Games and Economic Behavior, 44, 311–331 (2001)

    Article  MathSciNet  Google Scholar 

  58. Okamoto, Y.: Traveling Salesman Games with the Monge Property. Discrete Applied Mathematics, 138, 349–369 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  59. Potters, J., Curiel, I., Tijs, S.: Traveling Salesman Games. Mathematical Programming, 53, 199–211 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  60. Schmeidler, D.: The Nucleolus of a Characteristic Function Game. SIAM J. Appl. Math., 17, 1163 – 1170 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  61. Shapley, L.S.: A Value for n-Persons Games. In: Contributions to the theory of games II, Annals of Mathematical Studies 28. Princeton University Press, 307–317 (1953)

    Google Scholar 

  62. Shapley, L.S., Shubik, M.: The Assignment Game I: the Core. International Journal of Game Theory, 1, 111–130 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  63. Solymosi, T., Raghavan, T.E.S.: An Algorithm for Finding the Nucleolus of an Assignment Game. International Journal of Game Theory, 23, 119–143 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  64. Suijs, J.: Cost Allocation in Spanning Network Enterprises with Stochastic Connection Costs. Games and Economic Behavior, 42, 156–171 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  65. Tamir, A.: On the Core of Traveling Salesman Cost Allocation Games. Operations Research Letters, 8, 31–34 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  66. Tamir, A.: On the Core of Network Synthesis Games. Mathematical Programming, 50, 123–135 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  67. Tijs, S., Parthasarathy, T., Potters, J., Prasad, V.R.: Permutation Games: Another Class of Totally Balanced Games. OR Spektrum, 6, 119–123 (1984)

    Article  MATH  Google Scholar 

  68. Vidal, C., Goetschalckx, M.: A Global Supply Chain Model with Transfer Pricing and Transportation Cost Allocation. European Journal of Operational Research, 129, 134–158 (2001)

    Article  MATH  Google Scholar 

  69. Wang, Q., Parlar, M.: A Three-Person Game Theory Model Arising in Stochastic Inventory Control Theory. European Journal of Operational Research, 42, 1–21 (1994)

    Article  MathSciNet  Google Scholar 

  70. Woeginger, G.J.: On the Rate of Taxation in a Cooperative Bin Packing Game. Methods and Models of Operations Research, 42, 315–324 (1995)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Production Engineering and Management, Technical University of Crete, 73100, Sydney, Greece

    Yannis Marinakis & Athanasios Migdalas

  2. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

Authors
  1. Yannis Marinakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Athanasios Migdalas
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Panos M. Pardalos
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos , Altannar Chinchuluun  & Panos M. Pardalos ,  & 

  2. Department of Production Engineering and Management, Technical University of Crete, 73100, Greece

    Athanasios Migdalas

  3. Department of Production Engineering and Management, Aristotle University of Thessaloniki, Greece

    Leonidas Pitsoulis

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Marinakis, Y., Migdalas, A., Pardalos, P.M. (2008). Cost Allocation in Combinatorial Optimization Games. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds) Pareto Optimality, Game Theory And Equilibria. Springer Optimization and Its Applications, vol 17. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77247-9_9

Download citation

Publish with us

Policies and ethics

Search

Navigation